A group decision making method with preference analysis to re-build the Global Entrepreneurship Index

This study proposes a group decision making (GDM) method with preference analysis to re-build the Global Entrepreneurship Index (GEI). Specifically, a single decision maker is firstly identified using a specified individual judgement about the importance order of three sub-indices of the GEI. A preliminary group decision matrix is constructed in terms of taking all possible individual judgments into account. Then the analysis of the preferential differences and preferential priorities with respect to the preliminary group decision matrix is conducted to obtain a revised group decision matrix, in which preferential differences calculate the weighted differences as the degrees of differences among different alternatives for each decision maker, preferential priorities describe the favorite ranking of alternatives for each decision maker. Finally, we employ the Stochastic Multicriteria Acceptability Analysis for group decision-making (SMAA-2) to create the holistic acceptability indices for measuring the entrepreneurship performance. In addition, a satisfaction index is developed to indicate the merits of proposed GDM method. A case study using the GEI-2019 of 19 G20 countries is carried out to validate our GDM method.


Introduction
Entrepreneurship has come to be perceived as an engine of social and economic development throughout the world. Entrepreneurship is in particular defined as "the dynamic, institutionally embedded interaction between entrepreneurial attitudes, entrepreneurial abilities, and entrepreneurial aspirations by individuals, which drives the allocation of resources through the creation and operation of new ventures" [1,2]. This definition of entrepreneurship not only is driven by opportunity, but also regards the level of technology. The empirical evidence has indicated that entrepreneurial activity varies across stages of economic development, with a U-shaped relationship between the rate of entrepreneurship and level of development [1]. Similarly, Du et al. explore the impact of entrepreneurship on national the presence of multiple decision makers, we obtain a preliminary group decision matrix and thus explore its preference structures. More specifically, we study preferential difference and priorities for the evaluation of alternatives by individual decision makers in a decision group, in which preferential differences calculate the weighted differences as the degrees of differences among different alternatives for each decision maker, preferential priorities describe the favorite ranking of alternatives for each decision maker [7][8][9]. We modify the preliminary group decision matrix considering preferential differences and preferential priorities, and thereby make the use of Stochastic Multicriteria Acceptability Analysis for group decision-making (SMAA-2) to create a set of holistic acceptability indices for rebuilding the GEI [10,11].
The main purpose of this study is to re-design the GEI through the development of a GDM method with preference analysis. Compared to the existing literature innovating the methods of measuring entrepreneurial performance, our work contributes to this growing topic in the following aspects: 1. This article identifies decision makers using the individual judgments about the importance order among ATT, ABT, and ASP, and builds a preliminary group decision matrix by considering all possible individual judgments.

Entrepreneurship performance measurement
This study is most relevant to the literature on developing methods to measure entrepreneurship performance. Marcotte reviews and analyzes the existing entrepreneurship indices associated with their conceptual and methodological dimensions, which are further compared in 21 OECD countries [12]. Bonyadi and Sarreshtehdari conduct a critical review to assess the existing shortfalls in the model used for the computation of the GEI, especially for economies that have not participated in the GEM annual surveys [13]. Zoltán J. Á cs et al. provide a comprehensive comparison between the GEM dataset and the World Bank Group Entrepreneurship Survey (WBGES) dataset designed to measure entrepreneurship [14]. Zoltán J. Á cs et al. propose a novel concept of National Systems of Entrepreneurship and provide an approach to characterizing them [15]. Zoltán J. Á cs et al. develop a regional application of the GEI that captures the contextual features of entrepreneurship across regions in Spain, with the identification of weaknesses in the incentive structure that affect regional development [16]. Ibrahim et al. construct the Entrepreneurship Index as a measurement tool that effectively determines if one possesses the prospect of becoming a successful entrepreneur by assessing several essential aspects [17]. Song

GDM methods and applications
This work is also closely related to the literature on GDM. Keeney proposes a general normative model of group decision analysis based on a set of logical and operational assumptions similar to those used for individual decision analysis [21]. Melkonyan and Safra examine the random preference model and the possibility of violations of weak stochastic transitivity for the models with expected utility and betweenness like preferences [6]. Liang et al. develop a prospect theory-based method to fuse the individual preference approval structures in GDM [22]. Dong et al. provide a trust relationships consensus reaching process (CRP) with a feedback mechanism including two approaches of facilitating consensus reaching: 1) the leaderbased preference adjustment and 2) the trust relationships improvement [23]. Xu [28]. Tang and Liao summarize the challenges from conventional group decision making to LSGDM and present a state-of-the-art survey of main achievements in this field [29]. In addition to these theoretical innovations in GDM, there are many applications of GDM in different areas, for example, credit scoring [30], emergency decision support [31], e-procurement service provider selection Ramkumar and Jenamani [32], industrial robot selection [33], energy performance comparison [5], determining passenger demands and evaluating passenger satisfaction [34], sustainable building material selections [35], site selection of high-speed railway station [36], and among others.

Problem description
Released by The Global Entrepreneurship and Development Institute (The GEDI Institute), the GEI is a composite indicator that measures the quality and dynamics of entrepreneurship ecosystems at a national and regional level across the 3As: entrepreneurial attitudes, entrepreneurial abilities, and entrepreneurial aspirations. The entrepreneurship performance of nation/region i, i = 1, 2, . . ., m associate with sub-index j, j = 1, 2, 3 is represented by x ij , the value of which is usually normalized to mitigate adverse effects of data size using various regularization formulas, such as mean regularization, mix-max regularization, and Z-score regularization. In reality, trade-off justification through aggregation of equal weights is widely used. However, there are two-fold drawbacks with respect to the equal weighting scheme. First, this plausible decision completely ignores the fact that different decision makers may have different individual judgments about the importance of these sub-indices. Second, the common practice for an individual decision maker is to determine a set of weights with respect to each subindex w j , j = 1, 2, 3, in a subjective, or objective, or subjective-objective integrated manner. Determining weights actually play an important role in these multi-attribute performance evaluation problems. Appropriate weight elicitation methods can significantly increase the efficiency of the final decision. Numerous methods have been developed with significant advantages to obtain optimal weights. However, reaching consensus on exact weights even when the ranking is known remains a controversial issue. Therefore, it is extremely significant to comprehensively consider the wisdom of multiple decision makers. To the best of our knowledge, we are the first to re-build the GEI in such a GDM framework.

Methodology
We develop a GDM method with preference analysis for the general case in this section, which can be easily employed to re-build the GEI with three sub-indices, namely, Attitudes, Abilities and Aspiration. Consider the general situation in which the performance of entity i, i = 1, 2, . . ., m is calculated in terms of taking the arithmetic average of the scores attained on n subindices, x ij , i = 1, 2, . . ., m, j = 1, 2, . . ., n, that is, in which x ij are standardized data that can effectively alleviate the adverse impact of various PLOS ONE data magnitudes. The consideration of all possible importance sequence of sub-indices could deal with the drawbacks associated with the arithmetic average scheme [5]. The proposed GDM method consists of three phases. First, a preliminary group decision making matrix is formulated, in which a single decision maker is identified by an importance sequence of all sub-indices. Second, both the preference levels and preference priorities among entities for individual decision makers are analyzed to derive a revised group decision making matrix. Third, the SMAA-2 is used to implement the group decision aggregation.

Preliminary group decision making matrix
In line with Fu et al. [5], there are n! individual decision makers identified by extensively considering the importance orders among the sub-indices. For the ease of demonstration, we only investigate the scenario with w 1 �w 2 �. . .�w n , and w j , j = 1, 2, . . ., n is the importance degree of sub-index j. This scenario is reasonably identified as an individual decision maker and the decision results could be easily migrated to other decision makers. In this manner, the performance result for entity i can be determined by solving the following linear program: For the auxiliary parameters α j � 0, j = 1, 2, . . ., n, the ranked weights are defined as a j . This is consistent with given individual preference among sub-indices, Moreover, we define The linear program (2) therefore equals to the model below:

PLOS ONE
Let k _ 2 f1; 2; . . .; ng satisfies that s ik _ ¼ max s ik f g, the optimal solution to linear program (5) is then derived as As a consequence, the performance result of entity i with w 1 �w 2 �. . .�w n could be easily derived as the closed form: This scheme is simple-to-implement and easy-to-understand, and could be readily migrated to other situations. The whole procedure does not require using any linear optimizer and is implemented on the spreadsheet package without determining the precise values of weights. In the consideration of all possible importance sequence of sub-indices, we construct a preliminary group decision matrix with n! individual decision makers: where v p i , i = 1, 2, . . ., m, p = 1,2,. . ., n! denotes the score of entity i evaluated by individual decision maker p. The preference ranking of alternative i by decision maker p, is therefore determined by the v p i ; i ¼ 1; 2; . . . ; m; p ¼ 1; 2; . . . ; n!. However, the group decision aggregation using the preliminary group decision matrix is intuitive and may lack the investigation of the preference levels and preference priorities among entities for individual decision makers [5,7].

Revised group decision making matrix
In response to the aforementioned shortcomings associated with the preliminary group decision matrix, this section untangles and integrates the preferential differences and preferential priorities to capture the weight differences and alternative priorities for better aggregation of individual judgments with a collective commitment. Specifically, preferential differences calculate the weighted differences as the degrees of differences among different alternatives for each decision maker, preferential priorities describe the favorite ranking of alternatives for each decision maker.
The previous studies have claimed that each decision maker may have her/his own preferential differences among alternatives, which have been measured by means of developing several meaningful indices [7][8][9]. In accordance with Huang and Li [7], the weighted difference between alternatives i and t by decision maker p is defined as in which v p i denote the performance result of entity i evaluated by individual decision maker p. Since α pit only indicate the preferential differences for decision maker p, we need to normalize the mean differences for the n! decision makers to derive the preferential differences in the decision group. As a result, the weighted difference for decision maker p is obtained as in which α p is computed by taking the sum of α pit for m mÀ 1 ð Þ 2 times to consider all possible combinations of alternative pairs.
A small α p indicates that the preferential difference for decision maker p is small. This implies that the decision maker p is unable to or has no interest in decisively discriminating these alternatives, and may have similar preferences for all alternatives. In this sense, the weighted differences could be utilized as a means for an individual decision maker to highlight her/his preferential differences in the decision group.
In addition, decision makers would be in particular aware of whether their most preferred alternative is adopted in a group decision problem, and thus assign more importance to this alternative than to others [7][8][9]. In this regard, the preferential priority of alternative i by decision maker p is defined as b pi ¼ m φ pi , in which φ pi represents the preference ranking of alternative i by decision maker p. In the presence of tied alternatives with the identical ranking, they would be assigned with the ranking that is the average of their original ranking positions.
Then the preferential priorities of alternative i are aggregated as b i ¼ b pi , and would be normalized to obtain the preferential priority of alternative i in the decision group as Obviously, the preferential priority particularly gives more weight to the best ranked alternative, and thus is realistic and pragmatic for decision makers because it is able to indicate the extent to which decision makers expect their most favorite alternative to be adopted in a group decision problem.
In summary, based on the above preferential difference and preferential priority analysis, a revised group decision matrix is obtained below to make the consensus-and commitmentachieving in a decision group more realistic and reasonable.

Group decision aggregation using SMAA-2
As for the revised group decision matrix V * m�n! , this study proposes to employ SMAA-2 to aggregate alternative-wide performance. Relative to Stochastic Multicriteria Acceptability Analysis (SMAA), decision makers do not have to state their preferences explicitly or implicitly. SMAA examines the weight space and describes the score that puts each option in the preferred ranking position [11]. The rank acceptability indices are developed to measure the variety of difference preferences that support each alternative the best rank. SMAA-2 extends the analysis to the sets of weight vectors for any rank from best to worst for each decision alternative [10]. Finally, a set of holistic acceptability indices are formulated by integrating the rank acceptabilities using metaweights.

Preliminaries.
We examine a common GDM scenario with m alternatives that are assessed by n! group members, where neither the assessment values of group members nor the corresponding weights are precisely acknowledged, and a decision maker is responsible for consolidating the group decision. It is further assumed that the preference characteristic of the decision maker is expressed by a real-valued utility function u(x i , w), where a weight vector w quantifies the subjective preference of that decision maker. Imprecise or uncertain attribute values are indicated by random variables ξ is , using implicit or estimated joint probability distribution and density functions f(ξ) in the space X 2 R m×n! . Furthermore, decision maker's unknown or partially known preferences are represented by a weight distribution with density function f(w) in the set of feasible weights W, which is defined as Complete lack of knowledge about weights is represented in a "Bayesian" spirit by a uniform weight distribution in W. The distribution has the density function f w [10]. Then utility function is utilized to map stochastic criteria values and weight distributions into utility distributions u(ξ i , w).
As a result, SMAA determines for each alternative the set of favorable weights W i (ξ) as: Any weight w2W i (ξ) makes the utility of x i no less than the utility of any other alternatives. We develop the following ranking function to define the rank of each alternative as an integer from the best rank (= 1) to the worst rank (= m): in which ρ (false) = 0 and ρ (true) = 1. SMAA-2 is proposed based on analyzing the sets of favorable rank weight W r i x ð Þ, which are defined as: A weight w 2 W r i x ð Þ assigns utilities for the alternatives in this manner so that alternative x i obtains rank r.

Meaningful measures.
The first measure is the rank acceptability index b r i , which evaluates the variety of different valuations granting alternative x i rank r. The rank acceptability index is calculated as a multidimensional integral over the criteria distribution and the favorable rank weights using The rank acceptability indices can be illustrated graphically to investigate how different varieties of weights support each rank for each alternative. Obviously, the rank acceptability indices distribute in the range [0, 1]. b r i ¼ 1 implies that the alternative x i always achieve the rank r, no matter the choice of weights, and b r i ¼ 0 means that the alternative x i never obtain the rank r, irrespective to the determination of weights as well.
The weight space corresponding to the k best ranks (kbr) for an alternative can be described by means of the central kbr weight vector w k i , which is computed by The kbr confidence factor p k i is defined as the probability that the alternative receives any rank from 1 to k if the central kbr weight vector is selected, and is calculated as an integral over the criteria distribution in X by

Holistic acceptability index.
The problem of comparing the alternatives in terms of their rank acceptabilities is recognized as a "second-order" multicriteria decision problem [10]. This motivates the development of a complementary approach combining the rank acceptabilities to formulate holistic acceptability indices ah for each alternative: where θ r are so-called metaweights, satisfying 0 � θ m � θ m−1 �. . .� θ 2 � θ 1 � 1. Consequently, on the strength of the revised group decision matrix taking into account both preferential differences and preferential priorities, the SMAA-2 method is used to build the holistic acceptability indices for each alternative, which could be employed to implement the performance evaluation and comparison of all alternatives.

An empirical study
To validate our GDM method, an empirical study is conducted to re-build the Global Entrepreneurship Index using the data of GEI-2019, which reports the rankings of 137 countries/ regions, and provides confidence intervals for the GEI. Nineteen individual G20 countries are selected to illustrate the GDM method. The G20 consists of 19 individual countries and the European Union (EU). The EU is represented by the European Commission and the European Central Bank. Together, the G20 countries account for about 90% of global gross domestic product (GWP), 80% of world trade (or 75% if not traded within the EU), two-thirds of the world's population, and about half of the world's land area. The ATT, ABT, ASP and GEI raw data of the 19 individual economics of G20 are given in the Table 2 below, in which the countries are alphabetically listed.
In view of that the GEI is composed of ATT, ABT and ASP, we reasonably identify six (  Table 3. Moreover, Table 3 also reports the preferential differences α p and preferential priorities b G i . It is observed that decision makers TBS and SBT has the largest and smallest preferential differences as α TBS = 0.1746 and α SBT = 0.1585. This indicates that decision maker TBS has the largest interest in decisively discriminating these countries, and decision maker SBT has similar preference among these countries. In addition, the preferential priority of United States is largest (b G United States ¼ 0:2819). This implies that United States is most preferred for the group in terms of ranking. It is an intuitive decision result since all six decision makers put United States first on the table.
Based upon the preferential differences α p and preferential priorities b G i , the revised group decision matrix is obtained according to (11) and presented in Table 4 below. It is evident that all values have been modified to better reveal the preference structure of the group decision matrix.
In what follows, we apply the SMAA-2 to implement the group decision aggregation for the revised group decision matrix as   and holistic acceptability indices are reported in the following Table 5 and Fig 1. We observe that the rankings of Argentina, Australia, Brazil, Canada, France, Germany, Japan, South Africa, South Korea, Turkey, United Kingdom, and United States are unchanged, while other countries have multiple possibilities to be ranked at different ranking positions. 19 G20 countries are ranked in accordance with the obtained holistic acceptability indices, and are further compared with the ranking derived according to the GEI-2019. In addition, the rankings of ATT, ABT, and ASP are also reported for further comparisons, which are demonstrated in the Fig 2 below. We notice that 6 out of 19 countries are differently ranked between GEI and GDM, while the ranking positions of the rest countries are sufficiently robust and reliable. Additionally, most of the countries are ranked differently under the ATT, ABT, ASP, GEI and GDM, except for United States and Canada.
For the purpose of evaluating the effectiveness of our GDM method, we develop a satisfaction index to compare the degrees of satisfaction between the proposed GDM method (a h i ) and individual decision maker. The difference between the holistic acceptability level of an alternative derived by our GDM method and that evaluated by a single decision maker could be a reasonable measure of satisfaction, that is, The above satisfactory difference is normalized as , which has a conceptual interpretation by means of distance. In other words, the larger the satisfactory difference, the less satisfactory the alternative. Aside from the satisfactory difference, the differences between the rankings derived in accordance with a h i and any of other alternatives can also have a significant effect on the satisfactory levels of alternatives. Then the difference of preferential rankings is denoted as in which ρ ip indicates the preferential ranking of alternative i assessed by DM p, and p i represents the preferential ranking of alternative i in accordance with a h i . In harmony with Siskos et al.
[39], we therefore combine the above two satisfactory aspects to build the satisfaction index for DM p as below: According to the extant studies [7][8][9], the arithmetic average of κ p , p = 1, 2, . . ., 6 is utilized to represent the comprehensive satisfactory levels for all individual decision makers.
The simple additive weighting (SAW) method is usually recognized as common aggregation method for group decision, and is also known as the most widely-used method for real world group decision problems [9]. In this sense, the satisfaction indices of each decision maker are collected when using both SAW and our GDM methods, are reported in Table 6 below as well. We find that our GDM method could significantly increase the satisfactory levels of all decision makers. From the perspective of the decision group, the satisfactory level is improved as 0:9664À 0:7234 0:7234 � 100% ¼ 25:15%. Therefore, the proposed GDM method is more satisfactory than the SAW method in the group decision problem.

Conclusions
Complied and published by The GEDI Institute, the GEI is a composite indicator that measures the quality and dynamics of entrepreneurship ecosystems at a national and regional level across the 3As: ATT, ABT, and ASP. To deal with the drawbacks with respect to the arithmetic average aggregation scheme of the GEI, this study proposes to re-build the GEI by developing a GDM method with preference analysis. More specifically, an individual decision maker is firstly represented using a specified individual judgement about the importance order of various sub-indices of the GEI. A preliminary group decision matrix is therefore constructed by taking all possible individual judgments into account. The analysis about preferential differences and preferential priorities of the preliminary group decision matrix is the conducted to obtain a revised group decision matrix, in which preferential differences calculate the weighted differences as the degrees of differences among different alternatives for each decision maker, preferential priorities describe the favorite ranking of alternatives for each decision maker. We ultimately employ the Stochastic Multicriteria Acceptability Analysis for group decision-making (SMAA-2) to build the holistic acceptability indices for measuring the entrepreneurship performance. In addition, we create a satisfaction index to demonstrate the merits of proposed GDM method. An empirical study using the GEI-2019 of 19 G20 countries is carried out to show the implementation of our GDM method. Our GDM method has several remarkable advantages over the previous measures. First, the proposed GDM method allows us to capture the collective wisdom from a group decisionmaking perspective. Second, the data-driven mechanism of our method reduces the decision bias and enhances the result objectiveness. Third, the preference analysis and aggregation views allow for more detailed understanding of the entrepreneurship performance. This would help inform policymakers in improving the entrepreneurship in a suitable manner.
Meaningful implications could be derived from this study. First, various individual preferences among ATT, ABT, and ASP should be taken into account when constructing the GEI, while the arithmetic average aggregation scheme has several endogenous shortcomings. Second, the investigation of preference structure about the group decision matrix would provide more practical decision supports. Third, the revised GEI is accepted by all individual decision makers. In other words, the disparity of individual preferences could be appropriately handled in this study.
There are some meaningful future research directions for this research. First, the effectiveness of individual decision makers' decision results should be checked in the future research. For instance, the validity of closed-form solutions should be further verified. Second, apart from the preferential differences and preferential priorities for the group decision matrix, future research should investigate other aspects of preference analysis. Third, the merits of